Chapter 3 : Systems of Linear Equations Instructor Dr. Dang Van Vinh (11/2006)

1. Math Dept, Faculty of Applied Science, HCM University of Technology------------------------------------------------------------------------------------- Chapter 3: Systems of Linear Equations • Instructor Dr. Dang Van Vinh (11/2006)

2. CONTENTS---------------------------------------------------------------------------------------------------------------------------CONTENTS--------------------------------------------------------------------------------------------------------------------------- I –Systems of Linear Equations II – Homogeneous system

3. Definition of a System of Linear equations. A System of m linear equations in n unknowns has the form: I. Systems of Linear Equations--------------------------------------------------------------------------------------------------------------------------- The quantities a11, a12, …, amn are called the coefficients of the system. The quantities b1, b2, …, bm are called the free or constant terms of the systems.

4. Definition of Homogeneous System. Definition of Nonhomogeneous System. A system is called homogeneous if all its constant terms b1, b2, …, bm are equal to zero. A system is called nonhomogeneous if at least one of the constant terms b1, b2, …, bm is different from zero. I. Systems of Linear Equations--------------------------------------------------------------------------------------------------------------------------- The solution of system is such a collection n of the numbers c1, c2, …, cm which, being substituted into system for the unknowns x1, x2, …, xn, turns all the equations of the system into identities.

5. A system of linear equations has either: • no solution, • exactly one solution, or • infinitely many solutions inconsistent system consistent system Two linear systems are equivalent, if they have the same solution set. The basic strategy is to replace one system with an equivalent system that is easier to solve. I. Systems of Linear Equations---------------------------------------------------------------------------------------------------------------------------

6. Definition of elementary reduction operation An operation is called elementary reduction operation if it transforms one system to equivalent system. I. Systems of Linear Equations--------------------------------------------------------------------------------------------------------------------------- There are three elementary reduction operations. 1. Swapping: an equation is swapped with another. 2. Rescaling (multiplying by a scalar): an equation has both sides multiplied by a nonzero constant. 3. Pivoting: an equation is replaced by the sum of itself and a multiple of another. Remark: We need to prove that above three operations are elementary reduction operations.

7. Example Solve the system: I. Systems of Linear Equations--------------------------------------------------------------------------------------------------------------------------- Solution: x = 1; y = -1; z = 0

8. I. Systems of Linear Equations--------------------------------------------------------------------------------------------------------------------------- coefficient matrix: augmented matrix:

9. I. Systems of Linear Equations---------------------------------------------------------------------------------------------------------------------------

10. Definition of Basic and Free variable. The variable corresponding to pivot columns in the matrix are called basic variable. The other variable is called free variable. row operations I.System of Linear Equations x1, x3, x4: basic variables x2: free variables

11. If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set. The Kronecker Capelli Theorem If , then the system AX = b is inconsistent. If , then the system AX = b is consistent. If = number of unknowns , then the system AX = b has unique solution. If < number of unknowns , then the system AX = b has many solutions. I.System of Linear Equations

12. Using Row Reduction to solve a linear system I. System of Linear equations----------------------------------------------------------------------------------------------------------- 1. Write the augmented matrix of the system. 2. Use the row reduction algorithm to obtain an equivalent augmented matrix is echelon form. Decide whether the system is consistent. 3. Write the system of equations corresponding to the matrix 4. Rewrite each nonzero equation from step 3 so that its one basic variable is expressed in terms of any free variable appearing in the equation.

13. Example I. Systems of Linear Equations---------------------------------------------------------------------------------------------------------------------- The augmented matrix of a linear system has been transformed by a row operations into the form below. Determine if the system is consistent.

14. Example Solve the following system: I. Systems of Linear Equations--------------------------------------------------------------------------------------------------------------------

15. Example Solve the following system: I. Systems of Linear Equations---------------------------------------------------------------------------------------------------------------------------

16. Example Find the general solution of the linear system Basis variable: Free variable: General Solution: I. Systems of Linear Equation---------------------------------------------------------------------------------------------------------------------------

17. Example Find the general solutions of the systems whose augmented matrix is given as below I. Systems of Linear Equations---------------------------------------------------------------------------------------------------------------------------

18. Example Find the general solutions of the systems whose augmented matrix is given as below I. Systems of Linear Equations--------------------------------------------------------------------------------------------------------------

19. Example Find the general solutions of the systems whose augmented matrix is given as below I. Systems of Linear Equations-------------------------------------------------------------------------------------------------------------

20. Example Find the general solutions of the systems whose augmented matrix is given as below I. Systems of Linear Equations---------------------------------------------------------------------------------------------------------------------------

21. Example Determine the value(s) m such that the matrix is the augmented of a consistent linear system I. Systems of Linear Equations-------------------------------------------------------------------------------------------------------------

22. Example Determine the value(s) m such that the matrix is the augmented of a consistent linear system I. Systems of Linear Equations---------------------------------------------------------------------------------------------------------------------------

23. Example Determine the value(s) m such that the corresponding linear system has unique solution I. Systems of Linear Equations----------------------------------------------------------------------------------------------------------

24. Example Determine the value(s) m such that the corresponding linear system has unique solution I. Systems of Linear Equations---------------------------------------------------------------------------------------------------------------------------

25. Definition of Homogeneous System. A system is called homogeneous if all its constant terms b1, b2, …, bm are equal to zero. The homogeneous system always has zero solution x1 = x2 = … = xn = 0. This solution is called the trivial solution. The homogeneous system always possesses a unique solution – the trivial solution - if and only if rank (A) = n. II. Homogeneous Systems.---------------------------------------------------------------------------------------------------------------------------

26. The homogeneous system AX = 0 has nontrivial solution if and only if rank (A) < n. The homogeneous system AX = 0, where A is a square matrix, has nontrivial solution if and only if det(A) = 0. II. Homogeneous Systems.---------------------------------------------------------------------------------------------------------------------------

27. Example Determine the general solution for the following homogeneous system. II. Homogeneous Systems.---------------------------------------------------------------------------------------------------------------------------

28. Example Among all solutions that satisfy the homogeneous system Determine those also satisfy the nonlinear constraint y – xy = 2z II. Homogeneous Systems.---------------------------------------------------------------------------------------------------------------------------

29. Example If A is the coefficient matrix for a homogeneous system consisting of four equations in eight unknowns and if there are five free variables, what is rank (A)? Example Explain why a homogeneous system of m equations in n unknowns where m < n must always possess an infinite number of solutions. II. Homogeneous Systems.---------------------------------------------------------------------------------------------------------------------------

30. Example Determine the value(s) m such that the homogeneous system has nontrivial solution I. Systems of Linear Equations-------------------------------------------------------------------------------------------------------------